Properties

Label 2-756-252.139-c1-0-6
Degree $2$
Conductor $756$
Sign $-0.356 - 0.934i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (1.73 + i)4-s + (1.5 + 0.866i)5-s + (−2.59 + 0.5i)7-s + (−1.99 − 2i)8-s + (−1.73 − 1.73i)10-s + (−0.866 + 0.5i)11-s + (1.5 + 0.866i)13-s + (3.73 + 0.267i)14-s + (1.99 + 3.46i)16-s + 3.46i·17-s − 6.92·19-s + (1.73 + 3i)20-s + (1.36 − 0.366i)22-s + (4.33 + 2.5i)23-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.670 + 0.387i)5-s + (−0.981 + 0.188i)7-s + (−0.707 − 0.707i)8-s + (−0.547 − 0.547i)10-s + (−0.261 + 0.150i)11-s + (0.416 + 0.240i)13-s + (0.997 + 0.0716i)14-s + (0.499 + 0.866i)16-s + 0.840i·17-s − 1.58·19-s + (0.387 + 0.670i)20-s + (0.291 − 0.0780i)22-s + (0.902 + 0.521i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.356 - 0.934i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342879 + 0.497900i\)
\(L(\frac12)\) \(\approx\) \(0.342879 + 0.497900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 \)
7 \( 1 + (2.59 - 0.5i)T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 + (-4.33 - 2.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.33 - 7.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (7.5 + 4.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.59 + 4.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.9 - 7.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + (2.59 - 1.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.866 - 1.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.3iT - 89T^{2} \)
97 \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.47818335721524114851036407255, −9.853642201725904358848699900872, −8.894555827191252607500974925282, −8.387282796139102033383649574767, −6.92389502732016569875441528263, −6.58902975377835463624364613258, −5.56635856714428223629430058577, −3.81689470192943826932689519201, −2.79557132676329502095916191083, −1.69219405540070865016033986734, 0.39675497382747421589457457038, 2.03026746601353201996401001446, 3.20445848258021719601532402300, 4.84216476282678967518926252717, 6.03663304531640849620653186782, 6.49291423754908361695778921656, 7.57705733235018519250845117708, 8.521052972684259456327897723810, 9.312389230824993167828009151724, 9.854527429200666762836276365839

Graph of the $Z$-function along the critical line